Slogan: "rational numbers exist but do not take up space."
This is in a measure-theoretic sense, so literally 100% (not just 99.5%). On the number line, the measure of an interval is the (absolute) difference between its endpoints (the "length" of the interval). For any measurable set, you can approximate its size from above by covering its points with intervals and shrinking them as much as possible.
But because there are only countably many rational numbers, you can cover all of the rationals with a set of intervals with a finite total sum of lengths.
To be a bit more specific, if your rationals are r_1, r_2, r_3, ..., then for any small number e > 0, you can form the set of intervals
which have a total sum of lengths e/2 + e/4 + e/8 + ... = e and contain all of the rational numbers.
So for any positive number e > 0, the measure of the rational numbers is less than e, meaning that the measure of the rational numbers is 0.
To turn this into actual probabilities requires a little bit more work, since the measure of the number line, and hence that of the irrational numbers, is infinite, not "100%". But you could look at the probability of selecting a rational number in the interval [0, 1] and use the same reasoning as above to get a probability of 100% for an irrational number and 0% for a rational number.
(Edited to try to answer the objection more clearly.)
The intervals overlap heavily. Each of the intervals R_i contains infinitely many rational numbers, and all but finitely many of those will have corresponding intervals which are completely contained within the interval R_i. As a result, the estimate obtained by the construction described is always strictly smaller than the chosen e.
I appreciate your careful and patient explanation, and assay a superficial and snarky one of my own: it is impossible for the measure of the rationals, if it is to exist at all, to be exactly `e` for every small positive real number `e`. Even if we didn't know anything about the overlap of the intervals, we'd know at best that we had a (possibly non-strict) upper bound of `e`; but the only non-negative real number that can satisfy such a bound for every positive `e` is `0`.
The closest approximation I can find is pp. 292--293. (I don't have a copy and haven't read it in about twenty years, so there may be a better approximation available.)
Hadden and Ellie have a brief discussion about the potential to market dying in space as a "really nifty last indulgence", whereupon Hadden segues to the topic of immortality and says the following:
"Now, I'm not bringing this up so I can boast. I'm bringing it up for a practical reason. If we're figuring out ways to extend our lifespans, think of what those creatures on Vega must have done. They probably are immortal, or close enough. I'm a practical person, and I've thought a lot now about immortality. I've probably thought longer and more seriously about it than anyone else. And I can tell you one thing for sure about immortals: They're very careful. They don't leave things up to chance. They've invested too much effort in becoming immortal. I don't know what they look like, I don't know what they want from you, but if you ever get to see them, this is the only piece of practical advice I have for you: Something you think is dead cinch safe, they'll consider an unacceptable risk. If there's any negotiating you get to do up there, don't forget what I'm telling you."
Many libraries will purge checkout records once a patron returns books, but the increasing reliance on centralized cloud-based ILS vendors is making this harder to do reliably.
I'd appreciate it if you could name the online system your county uses. (Or if you can provide a link.)
Yes. Diaconis, Holmes, and Montgomery demonstrate (but do not provide a full specification for) a coin-flipping machine which will yield heads 100%* of the time (as long as it starts on heads).
Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.
EDIT: Just wanted to add that an order isomorphism has two requirements:
(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and
(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).
Thomas Pynchon's essay does a good job, I think, of explaining what the Luddites really fought against, and he does not, as this article seems to suggest, say a Luddite is "someone who opposes technological progress".
> But it's important to remember that the target even of the original assault of 1779, like many machines of the Industrial Revolution, was not a new piece of technology. The stocking-frame had been around since 1589 .... Now, given that kind of time span, it's just not easy to think of Ned Lud as a technophobic crazy.
> This International Standard specifies a datastream and an associated file format, Portable Network Graphics (PNG, pronounced "ping"), for a lossless, portable, compressed individual computer graphics image transmitted across the Internet.
According to the abstract at https://arxiv.org/abs/1008.1459 , this document has been revised many times (the current version, from 2015, is v38). Is there a particular reason that v8 was the version selected for submission?
> As we may have mentioned, due to the kindness of D. Jon Grossman's son, Jerome, we have the complete file of Jon's correspondence with Cummings. On making a preliminary tour through these letters, we found Jon preparing a French edition of his translations of Cummings' poetry, and on 27 February 1951 he wrote to the poet: "are you E.E.Cummings, ee cummings, or what?(so far as the title page is concerned)wd u like title page all in lowercase?"
> The poet replied on 1 March 1951: "E.E.Cummings, unless your printer prefers E. E. Cummings/ titlepage up to you;but may it not be tricksy svp[.]"
> That seems definitive to us: may it not be tricksy!
This is in a measure-theoretic sense, so literally 100% (not just 99.5%). On the number line, the measure of an interval is the (absolute) difference between its endpoints (the "length" of the interval). For any measurable set, you can approximate its size from above by covering its points with intervals and shrinking them as much as possible.
But because there are only countably many rational numbers, you can cover all of the rationals with a set of intervals with a finite total sum of lengths.
To be a bit more specific, if your rationals are r_1, r_2, r_3, ..., then for any small number e > 0, you can form the set of intervals
(r_1 - e/2^2, r_1 + e/2^2), (r_2 - e/2^3, r_2 + e/2^3), (r_3 - e/2^4, r_3 + e/2^4), ...
which have a total sum of lengths e/2 + e/4 + e/8 + ... = e and contain all of the rational numbers.
So for any positive number e > 0, the measure of the rational numbers is less than e, meaning that the measure of the rational numbers is 0.
To turn this into actual probabilities requires a little bit more work, since the measure of the number line, and hence that of the irrational numbers, is infinite, not "100%". But you could look at the probability of selecting a rational number in the interval [0, 1] and use the same reasoning as above to get a probability of 100% for an irrational number and 0% for a rational number.